The image inpainting problem consists of restoring an image from a (possibly noisy) observation, in which data from one or more regions are missing. Several inpainting models to perform this task have been developed, and although some of them perform reasonably well in certain types of images, quite a few issues are yet to be sorted out. For instance, if the image is expected to be smooth, the inpainting can be made with very good results by means of a Bayesian approach and a maximum a posteriori computation [2]. For non-smooth images, however, such an approach is far from being satisfactory. Even though the introduction of anisotropy by prior smooth gradient inpainting to the latter methodology is known to produce satisfactory results for slim missing regions [2], the quality of the restoration decays as the occluded regions widen. On the other hand, Total Variation (TV) inpainting models based on high order PDE diffusion equations can be used whenever edge restoration is a priority. More recently, the introduction of spatially variant conductivity coefficients on these models, such as in the case of Curvature-Driven Diffusion (CDD) [4], has allowed inpainted images with well defined edges and enhanced object connectivity. The CDD approach, nonetheless, is not quite suitable wherever the image is smooth, as it tends to produce piecewise constant restorations. In this work we present a two-step inpainting process. The first step consists of using a CDD inpainting to build a pilot image from which to infer a-priori structural information on the image gradient. The second step is inpainting the image by minimizing a mixed spatially variant anisotropic functional, whose weight and penalization directions are based upon the aforementioned pilot image. Results are presented along with comparison measures in order to illustrate the performance of this inpainting method.

The image inpainting problem consists of restoring an image from a (possibly noisy) observation, in which data from one or more regions are missing. Several inpainting models to perform this task have been developed, and although some of them perform reasonably well in certain types of images, quite a few issues are yet to be sorted out. For instance, if the image is expected to be smooth, the inpainting can be made with very good results by means of a Bayesian approach and a maximum a posteriori computation [2]. For non-smooth images, however, such an approach is far from being satisfactory. Even though the introduction of anisotropy by prior smooth gradient inpainting to the latter methodology is known to produce satisfactory results for slim missing regions [2], the quality of the restoration decays as the occluded regions widen. On the other hand, Total Variation (TV) inpainting models based on high order PDE diffusion equations can be used whenever edge restoration is a priority. More recently, the introduction of spatially variant conductivity coefficients on these models, such as in the case of Curvature-Driven Diffusion (CDD) [4], has allowed inpainted images with well defined edges and enhanced object connectivity. The CDD approach, nonetheless, is not quite suitable wherever the image is smooth, as it tends to produce piecewise constant restorations. In this work we present a two-step inpainting process. The first step consists of using a CDD inpainting to build a pilot image from which to infer a-priori structural information on the image gradient. The second step is inpainting the image by minimizing a mixed spatially variant anisotropic functional, whose weight and penalization directions are based upon the aforementioned pilot image. Results are presented along with comparison measures in order to illustrate the performance of this inpainting method.

During the last two decades several generalizations of the traditional Tikhonov-Phillips regularization method for solving inverse ill-posed problems have been proposed. Many of these variants consist essentially of modifications on the penalizing term, which force certain features in the obtained regularized solution ([11,18]). If it is known that the regularity of the exact solution is inhomogeneous it is often desirable the use of mixed, spatially adaptive methods ([7,12]). These methods are also highly suitable when the preservation of edges is an important issue, since they allow for the inclusion of anisotropic penalizers for border detection ([20]). In this work we propose the use of a penalizer resulting from the convex spatially-adaptive combination of a classic L2penalizer and an anisotropic bounded variation seminorm. Results on existence and uniqueness of minimizers of the corresponding Tikhonov-Phillips functional are presented. Results on the stability of those minimizers with respect to perturbations in the data, in the regularization parameter and in the operator are also established. Applications to image restoration problems are shown.

The success of machine learning algorithms strongly depends on the feature extraction and data representation stages. Classification and estimation of small repetitive signals masked by relatively large noise usually requires recording and processing several different realizations of the signal of interest. This is one of the main signal processing problems to solve when estimating or classifying P300 evoked potentials in brain-computer interfaces. To cope with this issue we propose a novel autoencoder variation, called Coherent Averaging Estimation Autoencoder with a new multiobjective cost function. We illustrate its use and analyze its performance in the problem of event related potentials processing. Experimental results showing the advantages of the proposed approach are finally presented.

We consider regularized solutions of linear inverse ill-posed problems obtained with generalized Tikhonov–Phillips functionals with penalizers given by linear combinations of seminorms induced by closed operators. Convergence of the regularized solutions is proved when the vector regularization rule approaches the origin through appropriate radial and differentiable paths. Characterizations of the limiting solutions are given. Finally, examples of image restoration using generalized Tikhonov–Phillips methods with convex combinations of seminorms are shown.

The obstructive sleep apnea?hypopnea (OSAH) syndrome is a very common and generally undiagnosed sleep disorder. It is caused by repeated events of partial or total obstruction of the upper airway while sleeping. This work introduces two novel approaches called most dicriminative activation selection (MDAS) and most discriminative column selection (MDCS) for the detection of apnea?hypopnea events using only pulse oximetry signals. These approaches use discriminative information of sparse representations of the signals to detect apnea?hypopnea events. Complete (CD) and overcomplete (OD) dictionaries, and three different strategies (FULL sparse representation, MDAS, and MDCS), are considered. Thus, six methods (FULL-OD, MDAS-OD, MDCS-OD, FULL-CD, MDAS-CD, and MDCS-CD) emerge. It is shown that MDCS-OD outperforms all the others methods. A receiver operating characteristic (ROC) curve analysis of this method shows an area under the curve of 0.937 and diagnostic sensitivity and specificity percentages of 85.65 and 85.92, respectively. This shows that sparse representation of pulse oximetry signals is a very valuable tool for estimating apnea?hypopnea indices. The implementation of the MDCS-OD method could be embedded into the oximeter so as to be used by primary attention clinical physicians in the search and detection of patients suspected of suffering from OSAH.

A brain computer interface (BCI) is a system which provides direct communication between the mind of a person and the outside world by using only brain activity (EEG). The event-related potential (ERP)-based BCI problem consists of a binary pattern recognition. Linear discriminant analysis (LDA) is widely used to solve this type of classification problems, but it fails when the number of features is large relative to the number of observations. In this work we propose a penalized version of the sparse discriminant analysis (SDA), called generalized sparse discriminant analysis (GSDA), for binary classification. This method inherits both the discriminative feature selection and classification properties of SDA and it also improves SDA performance through the addition of Kullback-Leibler class discrepancy information. The GSDA method is designed to automatically select the optimal regularization parameters. Numerical experiments with two real ERP-EEG datasets show that, on one hand, GSDA outperforms standard SDA in the sense of classification performance, sparsity and required computing time, and, on the other hand, it also yields better overall performances, compared to well-known ERP classification algorithms, for single-trial ERP classification when insufficient training samples are available. Hence, GSDA constitute a potential useful method for reducing the calibration times in ERP-based BCI systems.

The problem of restoring a signal or image is often tantamount to approximating the solution of a linear inverse ill-posed problem. In order to find such an approximation one might regularize the problem by penalizing variations on the estimated solution. Among the wide variety of methods available to perform this penalization, the most commonly used is the Tikhonov-Phillips regularization, which is appropriate when the sought signal or image is expected to be smooth, but it results unsuitable whenever preservation of discontinuities and edges is an important matter. Nonetheless, there are other methods with edge preserving properties, such as bounded variation (BV) regularization. However, these methods tend to produce piecewise constant solutions showing the so called “staircasing effect” and their numerical implementations entail great computational effort and cost. In order to overcome these obstacles, we consider a mixed weighted Tikhonov and anisotropic BV regularization method to obtain improved restorations and we use a half-quadratic approach to construct highly efficient numerical algorithms. Several numerical results in signal and image restoration problems are presented.

Several generalizations of the traditional Tikhonov-Phillips regularization method have been proposed during the last two decades. Many of these generalizations are based upon inducing stability throughout the use of different penalizers which allow the capturing of diverse properties of the exact solution (e.g. edges, discontinuities, borders, etc.). However, in some problems in which it is known that the regularity of the exact solution is heterogeneous and/or anisotropic, it is reasonable to think that a much better option could be the simultaneous use of two or more penalizers of different nature. Such is the case, for instance, in some image restoration problems in which preservation of edges, borders or discontinuities is an important matter. In this work we present some results on the simultaneous use of penalizers of L2 and of bounded variation (BV) type. For particular cases, existence and uniqueness results are proved. Open problems are discussed and results to signal restoration problems are presented.

We propose a model for the dynamics of an heterogeneous tumor, which consists of sensitive and resistant cells. The model is analyzed and validated using a cellular automaton whose local rules are classic and widely accepted in Biology. We then extend the model to a tumor under therapy. We consider Shannon’s entropy for the tumor and analyze the problem of minimizing this entropy. From this minimization problem, we find viable therapies that maintain at low level the entropy of the tumor. These therapies could provide a valuable tool for designing protocols for disease control, maintaining a very low growth level, while the tumor remains composed mainly of sensitive cells.